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Chapter 10 – Isoparametric Formulation. Isoparametric formulation is used for: 2-D non-rectangular quadrilateral elements (4 & 8 node) 3-D non-rectangular hexahedral (brick) elements (8 & 20 node) Commonly used in commerical codes Convenient for use with numerical integration

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chapter 10 isoparametric formulation
Chapter 10 – Isoparametric Formulation

Isoparametric formulation is used for:

  • 2-D non-rectangular quadrilateral elements (4 & 8 node)
  • 3-D non-rectangular hexahedral (brick) elements (8 & 20 node)

Commonly used in commerical codes

Convenient for use with numerical integration

Can be used with linear and higher order displacement interpolation functions

the term isoparametric
The term “Isoparametric”

“iso” – same

“parametric” – parameters

Isoparametric – “same parameters” are used to describe the displacement interpolation and the coordinate transformation

coordinate transformation

Coordinate Transformation

Global coordinate system

Natural coordinate system

isoparametric formulation applied to a bar element
Isoparametric Formulation applied to a Bar Element

Global Coordinate – x

Natural Coordinate - s

bar element coordinate transformation cont1
Bar Element Coordinate Transformation (cont.)

Recall bar element displacement interpolation functions:

Note: Same functions

element stiffness matrix
Element Stiffness Matrix

where

determinant of the Jacobian

chapter 10 isoparametric formulation cont
Chapter 10 – Isoparametric Formulation(cont.)

Today’s topics:

  • Numerical Integration (Gaussian Quadrature)
  • Evaluation of Stiffness Matrix using Gaussian Quadrature
  • Evaluation of Element Stresses
  • Higher order shape functions
rectangular plane stress element cont
Rectangular Plane Stress Element(cont.)

Assumed displacement interpolation – bilinear

In terms of nodal displacements

rectangular plane stress element cont1
Rectangular Plane Stress Element(cont.)

Displacement interpolation (matrix form)

where

rectangular plane stress element cont2
Rectangular Plane Stress Element(cont.)

Strain-displacement relation

Matrix form

Note: linear dependence on x & y

rectangular plane stress element cont3
Rectangular Plane Stress Element(cont.)

Element stiffness matrix

Element force matrix

Element equations

isoparametric coordinate transformation

Isoparametric Coordinate Transformation

Natural coordinate system

Global coordinate system

isoparametric coordinate transformation cont
Isoparametric Coordinate Transformation (cont.)

Coordinate transformation functions (same as displacment interpolation)

Matrix form

isoparametric element
Isoparametric Element

Element stiffness matrix

In terms of isoparametric coordinates

Need B(s,t) and determinant of Jacobian

isoparametric element cont1
Isoparametric Element (cont.)

Determinant of Jacobian (see text for details)

Nodal coordinates

isoparametric element cont3
Isoparametric Element (cont.)

Evaluation of [k]:

Requires numerical integration to evaluate double integral of the form:

gaussian quadrature

x

Gaussian Quadrature

Consider single integral of the form:

gaussian quadrature cont

Weight factor, W1= 2

x1= 0 is the sampling point

Gaussian Quadrature (cont.)

Approximate the integral by sampling the function at one point (n=1):

Note: result is exact if y(x) is a first order polynomial

gaussian quadrature n 2
Gaussian Quadrature (n=2)

Note: result is exact if y(x) is a third order polynomial

gaussian quadrature n 3
Gaussian Quadrature (n=3)

Note: result is exact if y(x) is a fifth order polynomial

example
Example

Exact solution:

double integral example
Double Integral - Example

Exact solution:

2.6613

evaluation of stiffness matrix cont

t

1

3

4

x

x

s

1

-1

1

2

x

x

-1

Evaluation of Stiffness Matrix (cont.)

For 4 node quad – 2 x 2 Full Integration

(Reduced Integration 1x1)

Gauss points or integration points

See text Example 10.4 for detailed example

evaluation of element stresses

3 x 8

8 x 1

3 x 3

3 x 1

Evaluation of Element Stresses
  • Options for computing stresses:
    • 1) Compute stresses at centroid (s = t = 0)
    • 2) Compute stresses at integration points
  • Extrapolate stress values to the nodes
  • No stress-averaging – plot color contours for each element
  • With stress-averaging – average stresses from adjacent elements at each node then plot color contours
higher order shape functions
Higher order shape functions

8 node quadratic isoparametric quad element

8 node isoparametric quad element cont

2 x 16

2 x 1

16 x 1

8 Node Isoparametric Quad Element (cont.)

Displacement interpolation:

element stiffness matrix1

=

3 x 16

3 x 3

16 x 3

16 x 16

3 x 3 Gaussian Quadrature – Full Integration

(2 x 2 – Reduced Integration)

Element Stiffness Matrix